Strong localization in seeded PageRank vectors

Transcription

1 Strong localization in seeded PageRank vectors David F. Gleich" Huda Nassar Computer Science" Computer Science " supported by NSF CAREER CCF , " IIS , DARPA SIMPLEX Kyle Kloster Mathematics "

2 Localization in seeded PageRank Newman s netscience graph 379 vertices 924 edges Inject dye here x is zero on most of the nodes

3 An example on a bigger graph Crawl of flickr from 2006: ~800K nodes, 6M edges," seeded PageRank with = error x true x nnz plot(x) x X-axis: node index Y-axis: value at that index in true PageRank vector nonzeros

4 Localization in seeded PageRank Given a seed and a graph What can we say about localization in the seeded PageRank vector with parameter? e s P = A T D 1 ( P)x = (1 )e s THEOREM We show that if the graph has a type of skewed degree dist. then the solution x cannot have many big entries. (Previously this was known only for const. degree or very slowly growing.)

5 Types of localization Weak (entry-wise)! kd 1 (x x )k 1 apple " Andersen, Chung, and Lang proved that the PageRank vector is weakly localized in the famous 2006 push paper. Strong (uniform)! kx x k 1 apple "

6 Strong localization When is strong localization possible? seed here Consider graphs with very slowly growing degree or a constant degree. An easy corollary of our subsequent theory. Also known from functions of sparse-matrix literature. Handles cases like the Erdős-Réyni graphs and grid graphs.

7 Strong localization can be impossible Consider a star graph 1 1+ (1 + )(n 1) Values in the PageRank vector seeded on the center node. Essentially everything is needed to be non-zero to get a global error bound.

8 Strong localization can be impossible Consider a star graph If we round k entries to zero, 1-norm error is so " " this:" " requires kx x k 1 apple " (1 + )" k (1 + )(n 1) (n 1) apple k 1 1+ (1 + )(n 1) Values in the PageRank vector seeded on the center node. Essentially everything is needed to be non-zero to get a global error bound.

9 Strong localization can be impossible Seeded PageRank is also non-local on" any complete bipartite graphs (generalizing star graphs)." " Why?" Fact: P is complete-bipartite iff eigenvalues = {-1,0,1}. PageRank is really a matrix function, f (x) = (1 x) 1.

10 Strong localization can be impossible Seeded PageRank is also non-local on" any complete bipartite graphs (generalizing star graphs)." " Why?" Fact: P is complete-bipartite iff eigenvalues = {-1,0,1}. PageRank is really a matrix function, f (x) = (1 x) 1. Fact: a matrix function is equiv to interpolating polynomial" " p( i )=f( i )! p(p) =f (P) " Only 3 eigenvalues p(x) is degree 2 (!) ( P) 1 e j = f (P)e j = (c 0 + c 1 P + c 2 P 2 )e j

11 When is localization possible? Graphs exist where" seeded PageRank has" no local behavior (star graphs) & graphs exist with" local behavior everywhere" ( degree <= constant, or log log(n) ) " So what properties can determine localization in seeded PageRank?

12 Skewed degree distributions The k-th largest degree d(k) apple max(dk p, ) 10 2 ( is min degree, p is decay exponent ) degree(k) 10 1 node id, k log log plot of the degree sequence for a synthetic example with 10,000 nodes d = 100 (max degree)" = 2 (min degree)" p = 0.5 (decay exponent) Distinct model from Pareto power law!

13 Strong localization in personalized PageRank Vectors Theorem (Nassar, K., Gleich): Let d be the max-degree, be the min-degree, n be the number of nodes, p be the decay exponent. Then the number of non-zeros N needed for kx x " k 1 apple " satisfies N apple min n, 1 C p (1/") 1 C p = ( d(1 + log d) p =1 d p (d (1/p) 1 1) otherwise Due to the maximum degree d, this does not say anything about traditional power-law graphs (e.g. the Pareto case)

14 Strong localization in personalized PageRank Vectors (sketch) We study the behavior of the " Gauss-Southwell or push algorithm for computing PageRank residual = remaining rank/dye to assign solution = assigned rank/dye Algorithm 1. pick node with most residual dye 2. assign dye to node 3. update residual dye on neighbors, 4. then repeat.

15 Strong localization in personalized PageRank Vectors (sketch) Define the residual vector r, r = (1 )e s (1 P)ˆx. Then kx ˆxk 1 < " is implied by krk 1 < "(1 ). Q k After k steps, kr k k 1 applekr 0 k 1 t=0 where Z (t) denotes the number of non-zero entries in r t. To guarantee kr k k 1 < "(1 ), it suffices to choose k so that, (( k + C p )/C p )( 1)/ apple " 1 1 Z (t) degree(k), The hard part is bounding Z (t) we show Z (t) apple C p + t race to min-degree! node id, k (The proof builds on techniques from [Gleich, K., Internet Math 2014] )

16 Asymptotic theory prediction α =0.25 n = n = 10 6 n = 10 7 n = 10 n = 10 8 n = 10 9 α =0.5 y-axis = number of non-zeros in approximate solution x-axis = 1/" red dashed line vector contains all non-zeros black line bound on non-zeros predicted by theorem

17 Asymptotic theory prediction α =0.25 n = n = 10 6 n = 10 7 n = 10 n = 10 8 n = 10 9 α =0.5 red dashed line vector contains all non-zeros black line bound on non-zeros predicted by theorem blue line actual number of non-zeros in approximation

18 Asymptotic theory prediction α =0.25 n = n = 10 6 n = 10 7 n = 10 n = 10 8 n = 10 9 α =0.5 red dashed line vector contains all non-zeros black line bound on non-zeros predicted by theorem blue line actual number of non-zeros in approximation L è Need a better bound

19 Empirical scaling guides a new bound varying α 10 2 varying p nonzeros(x ε ) / d log(d) 10 1 nonzeros(x ε ) / d log(d) /ε graph size, n = 10 6, d = p n At left, p = 0.95, black, green, blue, red, represent = {0.25, 0.3, 0.5, 0.65, 0.85} At right, = 0.5 and dashed blue, black and green represent p = {0.5, 0.75, 0.95} 1/ε

20 Empirical scaling guides a new bound varying α 10 2 varying p nonzeros(x ε ) / d log(d) 10 1 nonzeros(x ε ) / d log(d) /ε /ε We conjecture (bound ) nnz(x " ) apple d log(d) (1/4p 2 ) 1 "

21 Bound accurately predicts localization α =0.85 α =0.65 α =0.5 α =0.3 α =0.25 n = 10 4 n = 6 n = 10 n = 10 7 n = 10 8 n = 10 9

22 Conclusion and future work Examine broader classes of graphs empirically (like real-world networks)" Improve the localization theory to apply to a wider range of degree distributions" Explore other graphs without localization more specifically, relationship between diameter and localization Get a theorem for the Pareto power-law case!